αα-selfdecomposable distributions and related Ornstein–Uhlenbeck type processes

The concept of selfdecomposability has been generalized to that of αα-selfdecomposability, α∈Rα∈R, by many authors. We first mention the existing results on the class of αα-selfdecomposable distributions and investigate the remaining problems. We give complete characterizations by stochastic integrals with respect to Lévy processes for the case 1≤α<21≤α<2. The main topic of this paper is Langevin type equations and the corresponding Ornstein–Uhlenbeck type processes related to αα-selfdecomposable distributions.